Let \(\zeta ^{(n)}=\inf \lbrace t \ge 0: I^{(n)}_E(t)=0\rbrace \) , then the final size \(T^{(n)}\)  of the epidemic is given by
\(T^{(n)}=n-S^{(n)}(\zeta ^{(n)})\) . Let \(\tilde{X}^{(n)}=\lbrace \tilde{X}^{(n)}(t):t \ge 0\rbrace \) , where \(\tilde{X}^{(n)}(t)=(\tilde{S}^{(n)}(t), \tilde{I}^{(n)}_0(t), \tilde{I}^{(n)}_1(t), \dots ,\tilde{W}^{(n)}(t))\) , be the process having the same state space as \(X^{(n)}\) , viz. \(E^{(n)}\) , and jump rates \(\tilde{q}^{(n)}(\xi ,\xi ^{\prime })=q^{(n)}(\xi ,\xi ^{\prime })/(\lambda n^E)\)  \((\xi ,\xi ^{\prime } \in E^{(n)}, \xi \ne \xi ^{\prime })\) . Recall that \(n^E\)  is the total number of infective edges when corresponding to state \(\xi \) . The process \(\tilde{X}^{(n)}\)  stops when it reaches a state \(\xi \)  with \(n^E=0\) . For \(t \ge 0\) , let \(\tilde{I}^{(n)}_E(t)=\sum _{k=1}^{\infty } k \tilde{I}^{(n)}_k(t)\) . Let \(\tilde{\zeta }^{(n)}=\inf \lbrace t \ge 0: \tilde{I}^{(n)}_E(t)=0\rbrace \)  and \(\tilde{T}^{(n)}=n-\tilde{S}^{(n)}(\tilde{\zeta }^{(n)})\) . Note that \(\tilde{X}^{(n)}\)  is a random time-scale transformation of \(X^{(n)}\) , in which when \(X^{(n)}\)  is in state \(\xi \)  the clock for \(\tilde{X}^{(n)}\)  is slowed down by the factor \(\lambda n_E(\xi )\)  (i.e. by the total force of infection), so \(T^{(n)}\)  and \(\tilde{T}^{(n)}\)  have the same distribution. Consequently we can use the time-transformed process \(\tilde{X}^{(n)}\)  and \(\tilde{T}^{(n)}\)  to study the asymptotic behaviour of \(T^{(n)}\)  as \(n \rightarrow \infty \) . The advantage of using the time-transformed process is that, as \(n \rightarrow \infty \) , \(\tilde{\zeta }^{(n)}\stackrel{p}{\longrightarrow }\tilde{\zeta }<\infty \)  (where \(\tilde{\zeta }\)  is non-random and defined below), whereas \(\zeta ^{(n)}\stackrel{p}{\longrightarrow }\infty \) .
Such random time-scale transformations have a long history of application to the final outcome of epidemics; see, for example, [1]}, [2]}, page 467, [3]}, Section 5.2, and [4]}.
